Quantum correlations in two-fermion systems John Schliemann,

Quantum correlations in two-fermion systems

John Schliemann,¹J. Ignacio Cirac,²Marek Kus´,³Maciej Lewenstein,⁴ and Daniel Loss⁵

1Department of Physics, The University of Texas, Austin, Texas 78712

2Institut fu¨r Theoretische Physik, Universita¨t Innsbruck, A-6020 Innsbruck, Austria

3Centre for Theoretical Physics, Polish Academy of Sciences, 02668 Warsaw, Poland

4Institut fu¨r Theoretische Physik, Universita¨t Hannover, 30167 Hannover, Germany

5Department of Physics and Astronomy, University of Basel, CH-4056 Basel, Switzerland 共Received 18 December 2000; published 3 July 2001兲

We characterize and classify quantum correlations in two-fermion systems having 2K single-particle states.

For pure states we introduce the Slater decomposition and rank共in analogy to Schmidt decomposition and rank兲; i.e., we decompose the state into a combination of elementary Slater determinants formed by pairs of mutually orthogonal single-particle states. Mixed states can be characterized by their Slater number which is the minimal Slater rank required to generate them. For K⫽2 we give a necessary and sufficient condition for a state to have a Slater number 1. We introduce a correlation measure for mixed states which can be evaluated analytically for K⫽2. For higher K, we provide a method of constructing and optimizing Slater number witnesses, i.e., operators that detect Slater numbers for some states.

DOI: 10.1103/PhysRevA.64.022303 PACS number共s兲: 03.65.Ta, 89.70.⫹c I. INTRODUCTION

In recent years a lot of effort关1,2兴in quantum information theory共QIT兲has been devoted to the characterization of en- tanglement, which is one of the key features of quantum mechanics关3兴. The resources needed to implement a particu- lar protocol of quantum information processing 共see, e.g., 关4兴兲are closely linked to the entanglement properties of the states used in the protocol. In particular, entanglement lies at the heart of quantum computing关3兴. The most fundamental question with regard to entanglement is, given a state of a multiparty system, is it entangled or not 共i.e., is it separable 关5兴兲? If the answer is yes, then the next question is how strong the entanglement is. For pure states in bipartite sys- tems the latter question can be answered by looking at the Schmidt decomposition 关6兴, i.e., the decomposition of the vector in a product basis of the Hilbert space with a minimal number of terms. For mixed states already the first question is notoriously hard to answer. There exist, however, many separability criteria, such as the Peres-Horodecki criterion 关7,8兴 and more recent concepts such as entanglement wit- nesses and the corresponding ‘‘entanglement revealing’’

positive maps关9,10兴.

While entanglement plays an essential role in quantum communication between parties separated by macroscopic distances, the characterization of quantum correlations at short distances is also an open problem, which has received much less attention so far. In this case the indistinguishable character of the particles involved 共electrons, photons, etc.兲 has to be taken into account. In his classic book, Peres 关6兴 discussed the entanglement in elementary states of indistin- guishable particles. These are symmetrized or antisymme- trized product states for bosons and fermions, respectively. It is easy to see that all such states of two-fermion systems, and as well as such states formed by two noncollinear single- particle states in two-boson systems, are necessarily en- tangled in the usual sense. However, in the case of particles far apart from each other, this type of entanglement is not of

physical relevance: ‘‘No quantum prediction, referring to an atom located in our laboratory, is affected by the mere pres- ence of similar atoms in remote parts of the universe’’ 关6兴. This kind of entanglement between indistinguishable par- ticles being far apart from each other is not the subject of this paper. Our aim here is rather to classify and characterize the quantum correlations between indistinguishable particles 共in our case fermions兲at short distances. We discuss below why this problem is relevant for quantum information processing in various physical systems.

For indistinguishable particles a pure quantum state must be formulated in terms of Slater determinants or Slater per- manents for fermions and bosons, respectively. Generically, a Slater determinant contains correlations due to the exchange statistics of the indistinguishable fermions. As the simplest possible example, consider a wave function of two共spinless兲 fermions,

⌿共rជ1,rជ2兲⫽ 1

&关␾共rជ1兲␹共rជ2兲⫺␾共rជ2兲␹共rជ1兲兴, 共1兲 with two orthonormalized single-particle wave functions

␾^(rជ^{) and} ␹^(rជ). Operator matrix elements between such single Slater determinants contain terms due to the antisym- metrization of coordinates 共‘‘exchange contributions’’ in the language of Hartree-Fock theory兲. However, if the moduli of

␾^(rជ^{) and} ␹^(rជ) have only vanishingly small overlap, these exchange correlations will also tend to zero for any physi- cally meaningful operator. This situation is generically real- ized if the supports of the single-particle wave functions are essentially centered around locations being sufficiently apart from each other or the particles are separated by a suffi- ciently large energy barrier. In this case the antisymmetriza- tion present in Eq. 共1兲has no physical effect.

Such observations clearly justify the treatment of indistin- guishable particles separated by macroscopic distances as ef- fectively distinguishable objects. So far, research in quantum information theory has concentrated on this case, where the

1050-2947/2001/64共2兲/022303共9兲/$20.00 64 022303-1 ©2001 The American Physical Society

exchange statistics of particles forming quantum registers could be neglected or was not specified at all.

The situation is different if the particles constituting, say, qubits are close together and possibly coupled in some com- putational process. This the case for all proposals of quantum information processing based on quantum dot technology 关11–13兴. Here qubits are realized by the spins of electrons residing in a system of quantum dots. The electrons have the possibility of tunneling eventually from one dot to the other with a probability which can be modified by varying external parameters such as gate voltages and magnetic field. In such a situation the fermionic statistics of electrons is clearly es- sential.

Additional correlations in many-fermion systems arise if more than one Slater determinant is involved, i.e., if there is no single-particle basis such that a given state of N indistin- guishable fermions can be represented as an elementary Slater determinant 共i.e., fully antisymmetric combination of N orthogonal single-particle states兲. These correlations are the analog of quantum entanglement in separated systems and are essential for quantum information processing in non- separated systems.

As an example consider a ‘‘swap’’ process exchanging the spin states of electrons on coupled quantum dots by gating the tunneling amplitude between them 关12,13兴. Before the gate is turned on, the two electrons in the neighboring quan- tum dots are in a state represented by a simple Slater deter- minant and can be regarded as distinguishable since they are separated by a large energy barrier. When the barrier is low- ered, more complex correlations between the electrons due to the dynamics arise. Interestingly, as shown in Refs. 关12兴, 关13兴, during such a process the system must necessarily enter a highly correlated state that cannot be represented by a single Slater determinant. The final state of the gate opera- tion, however, is, similarly as the initial one, essentially given by a single Slater determinant. Moreover, by adjusting the gating time appropriately one can also perform a ‘‘square root of a swap’’ which turns a single Slater determinant into a ‘‘maximally’’ correlated state in much the same way 关13兴. At the end of such a process the electrons can again be viewed as effectively distinguishable, but are in a maximally entangled state in the usual sense of distinguishable sepa- rated particles. In this sense the highly correlated intermedi- ate state can be viewed as a resource for the production of entangled states.

We expect that similar scenarios apply to other schemes of quantum information processing that involve cold par- ticles 共bosons or fermions兲 interacting at microscopic dis- tances at which the quantum statistics becomes essential. For instance, it should be of relevance for quantum computing models employing ultracold atoms in optical lattices关14兴or ultracold atoms in arrays of optical microtraps 关15兴.

It is the purpose of the present paper to analyze the above type of quantum correlations between indistinguishable fer- mions in more detail. However, to avoid confusion with the existing literature we shall reserve in the following the term

‘‘entanglement’’ for separated systems and characterize the

analogous quantum correlation phenomenon in nonseparated fermionic systems by the notions of Slater rank and Slater number to be defined below.

We are going to formulate analogies with the theory of entanglement and translate several very recent results 关10,16,17兴 concerning standard systems of distinguishable parties (Alice⫽Bob) to the case of indistinguishable fermi- ons. In general, we will deal with a system of two fermions each of which live in a 2K-dimensional single-particle space.

The plan of the paper is as follows: In Sec. II we discuss pure states and formulate the analog of Schmidt decomposi- tion and rank—Slater decomposition and rank. We then dis- cuss a simple operational criterion for the case of two elec- trons in two neighboring quantum dots (K⫽2) to determine whether a given state is of Slater rank 1. This criterion was first derived in Ref.关13兴. In Sec. III we define the concept of a Slater number for mixed states. For K⫽2 we present the necessary and suficient condition for a mixed state to have the Slater number 1. This is an analog of the Peres- Horodecki criterion关7,8兴in the Wootters formulation关18兴. In Sec. IV we extend the results of Sec. III and define a Slater correlation measure which is the analog of the entanglement formation measure关19兴. This quantity can be calculated ana- lytically for the case K⫽2, in analogy to the Wootters result 关18兴. In Sec. V we turn to the case K⬎2 and introduce Slater number witnesses of canonical form 共defined in analogy to entanglement关9,16兴and Schmidt number关20,17兴witnesses兲. We construct examples of such k-Slater witnesses, which provide the necessary conditions for a given state to have a Slater number smaller than k; we also discuss optimization of Slater witnesses. Finally, we analyze the associated 关21兴 positive maps. We close by discussing further analogies, but also differences, between entanglement in separated systems of distinguishable particles as opposed to quantum correla- tions in nonseparated systems of indistinguishable particles.

II. SLATER RANK OF PURE STATES

We consider two indistinguishable fermions each of which resides in the single-particle Hilbert space C^2K. This situation is given, e.g., in a system of two electrons in K neighboring quantum dots where only the orbital ground state of each dot is taken into account. Alternatively, one may think of, say, two quantum dots with an appropriate number of orbital states available for the two fermions.

The states共density matrices兲in such a system are positive self-adjoint operators acting on the antisymmetric space A(C^2K丢C^2K). Let us first consider pure states, i.e., projec- tors on a vector 兩⌿典苸A(C^2K丢C^2K). Let f_a, f_a^†, a

⫽1, . . . ,2K, denote the fermionic annihilation and creation operators of single-particle states forming an orthonormal basis in C^2K, and兩⍀典denotes the vacuum state. Each vector in the two-electron space can be represented as 兩⌿典

⫽兺a,bw_abf_a^†f_b^†兩⍀典^{, where w}ab⫽⫺w_ba is an antisymmetric matrix. We have the following generalization of theorem 4.3.15 from Ref. 关22兴, which will allow us to define the fermionic analog of the Schmidt decomposition:

Lemma 1. For any antisymmetric N⫻N matrix A⫽0 there exists a unitary transformation U

⬘

such that A

⫽U

⬘

^ZU

⬘

^T, where the matrix Z has blocks on the diagonal,

Z⫽diag关Z₀,Z₁,...,Z_M兴, Z₀⫽0, Z_i⫽

冋

^{⫺0zi z0i}

册

^{, 共2兲}

and Z₀ is a (N⫺2 M )⫻(N⫺2 M ) null matrix.

Proof. Let A be an N⫻N, complex, antisymmetric matrix acting on C^N, A⫽⫺A^T; hence, A^†⫽⫺A*. Let us define B ªAA*_⫽⫺AA^†. Here B is Hermitian, B⫽B^†, and hence diagonalizable by a unitary transformation: B⫽UDU^†, UU^†⫽1, D-diagonal. Now consider CªU^†AU*. It is easy to check that C is antisymmetric, C^T⫽⫺C, and normal, CC^†⫽C^†C. Let us decompose C into its real and imaginary parts: C⫽F⫹iG; F, G are real N⫻N matrices. Since C is antisymmetric, so are F and G. Since C is normal, F and G commute. Thus F and G are real, antisymmetric, commuting matrices. Hence they can be simultaneously brought to block-diagonal forms by a real orthogonal transformation 关22兴, F⫽OF_bdO^T, G⫽OG_bdO^T, where O is an N⫻N ma- trix, OO^T⫽I, where

F_bd⫽diag关X₀,X₁,...,X_K兴,

G_bd⫽diag关Y₀,Y₁,...,Y_L兴, 共3兲 and X₀,Y₀ are null matrices of some dimensions, X₀⫽0, Y₀⫽0, whereas X_i,Y_i are standard antisymmetric 2⫻2 blocks:

X_i⫽

冋

^{⫺0xi x0i}

册

^{, Yi⫽}

冋

^{⫺0yi y0i}

册

^{. 共4兲}

Thus C⫽OZO^T where Z has the form 共2兲 and, finally A

⫽UCU^T⫽UOZO^TU^T⫽(UO)Z(UO)^T⫽U

⬘

^ZU

⬘

^{T with U}

⬘

unitary.䊏

Lemma 2. Every vector in antisymmetric space A(C^2K丢C^2K) can be represented in an appropriately chosen basis inC^2K in a form of the Slater decomposition

兩⌿典⫽ 1

冑

兺i⫽1 K 兩z_i兩²i

兺

⫽1

K

z_if_a

1共i兲

† f_a

2共i兲

† 兩⍀典^, 共5兲

where the states f_a

1(i)

† 兩⍀典^{, f}a₂(i)

† 兩⍀典^{, i}⫽1, . . . ,K, form an orthonormal basis in C^2K; i.e., each of these single-particle states occurs only in one term in the summation 共5兲. The number of nonvanishing coefficients z_i 共i.e., the number of elementary Slater determinants required to construct兩⌿典兲 is called the Slater rank.

Proof. Let 兩⌿典⫽兺a,bw_abf_a^†f_b^†兩⍀典. Note that the change of basis in C^2K corresponds to a unitary transformation of fermionic operators, f_a^†⫽兺bU_ba( f_b

⬘

^)†, which implies that in the new basis w

⬘

⫽UwU^T. From lemma 1 we may choose U such that w

⬘

will have the form 共2兲, which provides the Slater decomposition.䊏

From the point of view of applications in quantum dot computers, it is important to be able to distinguish states with Slater rank 1 共which can be easily prepared and detected兲 from those that involve more than one elementary Slater de-

terminant. In general, given 兩⌿典 in some basis, in order to check the Slater rank, one has to perform the Slater decom- position. As we know from Ref.关13兴, the situation is simpler for the case K⫽2, where we have the following.

Lemma 3 共Ref. 关13兴兲. A vector 兩⌿典^⫽兺a,b⫽1

4 w_abf_a^†f_b^†兩⍀典 inA(C⁴丢C⁴) has Slater rank 1 iff

␩共兩⌿典⁾⫽

冏

^a,b,c,d

^兺

^{⑀abcdwabwcd}

冏

^{⫽0, 共6兲}

where ⑀^abcd denotes the totally antisymmetric tensor in C⁴丢C⁴丢C⁴丢C⁴.

Remark. The quantity␩共兩⌿典兲can be constructed from the dual state

兩⌿˜典^⫽

兺

_a,b ^{w˜abfa†fb†兩⍀典, 共7兲}

defined by the dual matrix

w

˜_ab⫽1

2

兺

_c,d ^{⑀abcdwcd* . 共8兲}

With these definitions we have

␩共兩⍀典^)⫽兩具^⌿˜兩⌿典^{兩. 共9兲} The proof of this lemma was presented first in Ref.关13兴. An alternative proof can be given using lemma 1 and observing that

det w⫽

冉

^{18具⌿˜兩⌿典}

冊

^{2, 共10兲}

where w is the antisymmetric 4⫻4 matrix defining兩⌿典. In the Appendix we list some further useful properties of

␩共兩⌿典兲 and the relation of the dualization operation to an antiunitary implementation of particle-hole transformation.

An interesting further question is possible generalizations of the above result to the case of K fermions having a single- particle space C^2K.

III. SLATER NUMBER OF MIXED STATES Let us now generalize the concepts introduced above to the case of mixed states. To this end, we define the Slater number of a mixed state, in analogy to the Schmidt number for the case of distinguishable parties 关20,17兴

Definition 1. Consider a density matrix␳of a two-fermion system and all its possible convex decompositions in terms of pure states, i.e., ␳⫽兺ip_i兩␺_i^ri典具␺_i^ri兩, where r_i denotes the Slater rank of兩␺_i^ri典; the Slater number of␳, k, is defined as k⫽min兵^rmax其^{, where rmax}is the maximum Slater rank within a decomposition, and the minimum is taken over all decom- positions.

In other words, k is the minimal Slater rank of the pure states that are needed in order to construct␳, and there is a construction of ␳ that uses pure states with Slater rank not exceeding k.

Many of the results concerning Schmidt numbers can be transferred directly to the Slater number. For instance, let us denote the whole space of density matrices in A(C^2K丢C^2K) by Sl_Kand the set of density matrices that have Slater num- ber k or less by Sl_k. Here Sl_kis a convex compact subset of Sl_K; a state from Sl_kwill be called a state of共Slater兲class k.

Sets of increasing Slater number are embedded into each other, i.e., Sl₁傺Sl₂傺¯Sl_k¯傺Sl_K. In particular, Sl₁ is the set of states that can be written as a convex combination of elementary Slater determinants; Sl₂ is the set of states of Slater number 2, i.e., those that require at least one pure state of Slater rank 2 for their formation, etc.

The determination of the Slater number of a given state is in general a very difficult task. Similarly, however, as in the case of separability of mixed states of two qubits共i.e., states in C²丢C²兲 and one qubit and one qutrit 共i.e., states in C²丢C³兲 关8兴, the situation is particularly simple in the case of small K. For K⫽1 there exists only one state共a singlet兲. For K⫽2 we will present below a necessary and sufficient con- dition for a given mixed state to have a Slater number of 1.

One should note, however, that in the considered case of fermionic states there exists no simple analogy of the partial transposition, which is essential for the theory of entangled states. In fact, the Peres-Horodecki criterion 关7,8兴 in 2⫻2 and 2⫻3 spaces says that a state is separable iff its partial transpose is positive. It is known, however, that the Peres- Horodecki criterion is equivalent to Wootters’ result 关18兴, relating separability to a quantity called concurrence, which is related to eigenvalues of a certain matrix. This latter ap- proach can be used to characterize fermionic states inA(C⁴

丢C⁴). We have the following theorem.

Theorem 1. Let the mixed state acting inA(C⁴丢C⁴) have a spectral decomposition ␳⫽兺i⫽1

r 兩⌿i典具⌿i兩, where r is the rank of ␳, and the eigenvectors 兩⌿i典 belonging to nonzero eigenvalues ␭i are normalized as 具^⌿i兩⌿j典^⫽␭i␦i j. Let 兩⌿i典^⫽兺a,bw_abⁱ f_a^†f_b^†兩⍀典 in some basis and define the com- plex symmetric r⫻r matrix C by

C_{i j}⫽_abcd

兺

^{⑀abcdwabi wcdj , 共11兲}

which can be represented using a unitary matrix as C

⫽UC_dU^T, with C_d⫽diag关c₁,c₂,¯,c_r兴 diagonal and 兩c₁兩

⭓兩c₂兩⭓¯⭓兩c_r兩. The state␳ has Slater number 1 iff

兩c₁兩⭐i

兺

⫽2 r

兩c_i兩. 共12兲

Proof. Let us assume that a state ␳ ^{acting in} A(C⁴丢C⁴) has Slater number 1, i.e.,

␳⫽

兺

_i⫽^r₁ ^{兩⌿i典具⌿i兩⫽}_k

兺

_⫽^r⬘₁ ^{兩␾k典具␾k兩, 共13兲}

where all ␾khave Slater rank 1, whereas r

⬘

can be an arbi- trary integer ⭓r. But 兩␾k典 can be represented as 兩␾k典

⫽兺i⫽1

r U_ki兩⌿i典⫽兺i⫽1

r 兺a,bU_kiw_abⁱ f_a^†f_b^†兩⍀典. From lemma 3

we obtain that for each k, ␩„w

⬘

^(k)…⫽0, where w

⬘

^(k)ab

⫽兺i⫽1

r U_kiw_abⁱ . The matrices U_ki must therefore fulfill, for every k,

abcd

兺 兺

_j⫽^r₁ ^{⑀abcdwabi wcdj UkiUk j⫽}_{i, j}

兺

_⫽^r ₁ ^{Ci jUkiUk j⫽0.}

共14兲 On the other hand, from Eq.共13兲we obtain

k

兺

⫽1 r⬘

U_kiU_{k j}*_⫽␦i j. 共15兲 The Slater rank 1 is thus equivalent to the existence of the r

⬘

^{⫻r matrix U}ki that fulfills Eqs.共14兲and共15兲. It is conve- nient to represent the rows of the matrix U_ki as vectors兩R_k典 in an r dimensional Hilbert space Haux. Equations共14兲and 共15兲 then reduce to 兺k

r⬘兩R_k典具^Rk兩⫽1, and具^Rk*_兩C兩R_k典⫽0 for all k. One can always change the basis inHaux, i.e., replace 兩R_k典^→U兩Rk典. Such a transformation does not affect Eq.共15兲 and transforms C→U^TCU. Since C is symmetric, U can be chosen in such a way that U^TCU is diagonal, and Eq. 共14兲 reads then兺i⫽1

r c_iU_ki²⫽0. In this new basis the construction of U_kiusing the method of Wootters关18兴can be carried over.

One can always assume that c₁U_k1² is real and positive by chosing the phases of兩R_k典. Then one observes that, provided Eq. 共14兲is fulfilled,

0⫽

冏

ⁱ

^兺

^{⫽r1 ciUki2}

冏

^{⭓兩ci兩兩Uk12 兩⫺i}

^兺

^{⫽r2兩ci兩兩Uki2兩. 共16兲}

Summing the above inequality over k and using Eq.共15兲, we obtain the necessary condition

兩c₁兩⭐_i

兺

_⫽^r₂ ^{兩ci兩. 共17兲}

To show that it is also a sufficient condition, we take r

⬘

^{⫽2 if} r⫽2, r

⬘

⫽4 if r⫽3,4, r

⬘

⫽8 if r⫽5,6, and U_ki⫽

⫾1_kiexp(i␪i)/

冑

^r

⬘

. The equations in Eq. 共14兲 are then all equivalent to

兩c₁兩⫽_i

兺

_⫽^r₂ ^{ciexp共2i␪i兲, 共18兲}

and the angles ␪i can indeed be chosen to assure that Eq.

共18兲is fulfilled, provided the condition共17兲holds. The⫾1_ki signs are designed in such a way that Eq. 共15兲 is fulfilled.

Thus for r

⬘

⫽2 we take共⫹⫹兲, 共⫹⫺兲for i⫽1,2, for r

⬘

⫽4 we take 共⫹⫹⫹⫹兲, 共⫹⫹⫺⫺兲, 共⫹⫺⫹⫺兲, 共⫹⫺⫺⫹兲 for i

⫽1, . . . ,4共or any 3 of them for i⫽1, . . . ,3兲, and finally for

r

⬘

⫽8, 共⫹⫹⫹⫹⫹⫹⫹⫹兲, 共⫹⫹⫹⫹⫺⫺⫺⫺兲,

共⫹⫹⫺⫺⫹⫹⫺⫺兲, 共⫹⫹⫺⫺⫺⫺⫹⫹兲, 共⫹⫺⫹⫺⫹⫺⫹⫺兲, 共⫹⫺⫹⫺⫺⫹⫺⫹兲. In the latter case we take again as many vectors as we need, i.e., i⫽1, . . . ,5⭐r⭐6.

The above theorem is an analog of the Peres-Horodecki- Wootters result for two-fermion systems having a single- particle space of dimension 2K⭐4. The situation is much

more complicated when we go to K⬎2; this is similar to the case of the separability problem in C^M丢C^N with M N⬎6.

These issues are investigated in Sec. V. In the following sec- tion, however, we shall concentrate on the case K⫽2.

IV. SLATER CORRELATION MEASURE

The similarity of our approach to that of Wootters 关18兴 can be pushed further and, in particular, allows us to define and calculate, for the case of K⫽2, the ‘‘Slater formation measure’’ 共in analogy to entanglement formation measure 关19兴兲.

To this aim we first consider a pure 共normalized兲 state 兩␺^¯典^⫽兺a,bw_abf_a^†f_b^†兩⍀典and define the Slater correlation mea- sure of兩␺^¯典 as in lemma 3共cf. Ref.关13兴兲,

␩共兩␺^¯典⁾⫽兩具␺ន_兩␺^¯典兩, 共19兲 with兩␺ន典being the dual of兩␺^¯典. Obviously, the notion of dual states, as well as the function␩共•兲in Eq.共19兲, can be defined also for unnormalized states. In the following we will denote such unnormalized states just as states occurring in the pre- vious sections, i.e., without the overbar.

The measure共19兲has all desired properties关19,23兴, such that it vanishes iff 兩␺^¯典 has Slater rank 1 and it is invariant with respect to local bilateral unitary operations or, in an- other words, with respect to changes of the basis in the single-particle space.

Having defined the measure for the pure states, we can consider the following definition

Definition 2. Consider a density matrix ␳ ^{acting in} A(C⁴丢C⁴) and all its possible convex decompositions in terms of pure states, i.e., ␳⫽兺i兩␺i典具␺i兩⫽兺ip_i兩␺^¯i典具␺^¯i兩, where the unnormalized states兩␺i典^⫽

冑

^pi兩␺^¯i典; the Slater cor- relation measure of␳^,CSl(␳), is defined as

CSl共␳兲⫽inf

再 ^兺

^{i pi␩共兩␺¯i典)}

冎

^,

where the infimum is taken over all decompositions.

In other words, CSl(␳) is the minimal amount of Slater correlations of the pure states that are needed in order to construct ␳, and there is a construction of␳ that uses pure states with ‘‘averaged’’ Slater correlation CSl(␳^).

Note that 兺ip_i␩⁽兩␺^¯i典^)⫽兺i␩⁽兩␺i典). As we shall see be- low, the measureCSl(␳) can be related directly to the matrix C_{i j} in Eq. 共11兲and to its ‘‘concurrence.’’ It is invariant not only with respect to local bilateral unitary operations, but it also cannot increase under local bilateral operations. These are trace preserving maps of the form ␳→M (␳⁾

⫽兺jA_j丢A_j␳^Aj

†丢A^†_j, where each A_j acts in C⁴, and 兺jA^†_jA_j丢A_j^†A_j⫽1. Such transformations correspond to mix- tures of density matrices obtained after nonunitary changes of the basis in the single-particle space. It is easy to see that

CSl„M共␳兲…⫽

冉 ^兺

^{j 兩det Ai兩CSl共␳兲}

冊

^{⭐CSl共␳兲.}

We have the following theorem

Theorem 2. For any␳ ^{acting in}A(C⁴丢C⁴),

CSl共␳兲⫽兩c₁兩⫺

兺

_i⫽^r⬘₂ ^{兩ci兩, 共20兲}

where c_i are the diagonal elements of C [Eq. (11)] in the basis that diagonalizes it.

Proof. The proof is essentially the same as the one in the previous section. Let us consider an arbitrary expansion of a given density matrix, ␳⫽兺k⫽1

r⬘ 兩␾k典具␾k兩, where 兩␾k典

⫽兺j⫽1

r U_{k j}兩⌿j典^{. Here 兩⌿}j典 denote the usual ‘‘subnormal- ized’’ eigenvectors of␳^with具^⌿j兩⌿j典being equal to the j th nonzero eigenvalue of ␳关18兴. It is easy to see that

CSl共兩␾k典具␾k兩兲⫽

冏

^{i, j}

^兺

^{r⫽1 Ci jUkiUk j}

冏

^{, 共21兲}

and 兺k⫽1

r⬘ U_ki*U_{k j}⫽␦i j. By changing the basis to the one in which C is diagonal, we get 共after choosing the phases of U_k1 such that c₁U_k1² are real and positive兲

k

兺

⫽1 r⬘

CSl共兩␾k典具␾k兩兲⫽k

兺

⫽1 r⬘

冏 ^兺

^{j cjUk j2}

冏

^{⭓兩c1兩⫺i}

^兺

^{⫽r⬘2 兩ci兩.}

共22兲 This inequality becomes an equality when we use the same construction of U_{k j} as in previous section, namely, U_{k j}⫽

⫾1_{k j}exp(i␪j)/

冑

^r

⬘

^{, with} ␪j selected in such a way that 共in- dependently of k兲

冏 ^兺

^{j cjUk j2}

冏

^⫽r1

^⬘ 冉

^兩c1兩⫺i

^兺

^{⫽r⬘2兩ci兩}

冊

^{. 共23兲}

䊏

The above construction provides, to our knowledge, a rare example of an analog of the entanglement formation measure that can be evaluated analytically. Obviously, since we have introduced the concept of Slater coefficients, we may define other Slater correlations measures for pure states in terms of appropriately designed convex functions of the Slater coeffi- cients 共in analogy to entanglement monotones 关24兴兲. For K

⫽2 and most probably only for K⫽2, all those measures are equivalent and the corresponding induced measures for mixed states can be calculated analytically.

V. SLATER WITNESSES

We now investigate fermion systems with single-particle Hilbert spaces of dimension 2K⬎4. In this case, a full and explicit characterization of pure and mixed state quantum correlations, such as given above for the two-fermion system with K⫽2, is apparently not possible. Therefore one has to formulate other methods to investigate the Slater number of a given state. We can, however, follow here the lines of the papers that we have written on entanglement witnesses 关10,16兴and Schmidt number witnesses关17兴.

In order to determine the Slater number of a density ma-

trix␳, we note that due to the fact that the sets Sl_kare convex and compact, any density matrix of class k can be decom- posed into a convex combination of a density matrix of class k⫺1, and a remainder␦关25兴.

Proposition 1. Any state of class k,␳k, can be written as a convex combination of a density matrix of class k⫺1 and a so-called k-edge state␦^:

␳k⫽共1⫺p兲␳k⫺1⫹p␦^{, 1}⭓p⬎0, 共24兲 where the edge state ␦has Slater number⭓k.

The decomposition共24兲is obtained by subtracting projec- tors onto pure states of Slater rank smaller than k, P

⫽兩␺^⬍k典具␺^⬍k兩, such that␳k⫺␭P⭓0. Here兩␺^⬍k典 ^{stands for} pure states of Slater rank r⬍k. Denoting by K(␳^{), R(}␳^), and r(␳) the kernel, range, and rank of␳, respectively, we observe that ␳

⬘

⬀␳⫺␭兩␺^⬍k典具␺^⬍k兩 is non negative iff兩␺^⬍k典 苸R(␳^{) and} ␭⭐具␺^⬍k兩␳^⫺1兩␺^⬍k典^{⫺1 共see 关25兴兲}. The idea be- hind this decomposition is that the edge state ␦ ^{which has} generically lower rank contains all the information concern- ing the Slater number k of the density matrix␳k.

As in the case of Schmidt number, there is an optimal decomposition of the form 共24兲 with p minimal. Alterna- tively, restricting ourselves to decompositions ␳k

⫽兺ip_i兩␺_i^ri典具␺_i^ri兩with all r_i⭐k, we can always find a decom- position of the form 共24兲 with ␦苸Sl_k. We define below more precisely what an edge state is.

Definition 3. A k-edge state ␦ is a state such that ␦

⫺⑀兩␺^⬍k典具␺^⬍k兩 is not positive, for any⑀⬎0 and兩␺^⬍k典^. Criterion 1. A mixed state ␦ is a k-edge state iff there exists no兩␺^⬍k典 ^{such that}兩␺^⬍k典苸R(␦^).

Now we are in the position of defining a k-class Slater witness共k-SW, k⭓2兲:

Definition 4. A Hermitian operator W is a Slater witness 共SW兲 of class k iff Tr(W␴⁾⭓0 for all ␴苸Sl_k⫺1 and there exists at least one␳苸Sl_k such that Tr(W␳⁾⬍0.

It is straightforward to see that every SW that detects ␳ given by Eq. 共24兲 also detects the edge state ␦^{, since if} Tr(W␳⁾⬍0, then necessarily Tr(W␦⁾⬍0, too. Thus knowl- edge of all SW’s of k-edge states fully characterizes all ␳ 苸Sl_k. Below, we show how to construct for any edge state a SW which detects it. Most of the technical proofs used to construct and optimize Slater witnesses are very similar to those presented in Ref.关10兴for entanglement witnesses.

All the operators we consider below act in A(C^2K丢C^2K). Let␦be a k-edge state, C an arbitrary positive operator such that Tr(␦^C)⬎0, and P a positive operator whose range fulfills R( P)⫽K(␦^{). We define} ⑀

⬅inf_兩␺⬍k典具␺^⬍k兩P兩␺^⬍k典 ^{and c⬅sup}具␺兩C兩␺典. Note that c

⬎0 by construction and ⑀⬎0, because R( P)⫽K(␦^{), and} therefore, since R(␦) does not contain any 兩␺^⬍k典 ^{by the} definition of edge state, K( P) cannot contain any兩␺^⬍k典 ^ei- ther. This implies the following.

Lemma 4. Given a k-edge state␦^{, then} W⫽P⫺⑀

cC 共25兲

is a k-SW which detects ␦^.

The simplest choice of P and C consists in taking projec- tions onto K(␦) and the identity operator on the asymmetric space 1^a, respectively. As we will see below, this choice provides us with a canonical form of a k-SW.

Proposition 2. Any Slater witness can be written it the canonical form

W⫽W˜⫺⑀1^a, 共26兲 such that R(W˜ )⫽K(␦^{), where}␦is a k-edge state and 0⬍⑀

⭐inf_{兩␺典苸S}

k⫺1具␺兩W˜兩␺典^.

Proof. Assume W is an arbitrary k-SW such that Tr(W␴⁾⭓0 for all␴苸Sl_k⫺1 and there᭚at least one␳^such that Tr(W␳⁾⬍0. Here W has at least one negative eigen- value. Construct W⫹⑀1a⫽W˜ , such that W˜ is a positive op- erator on A(C^2K丢C^2K), but does not have a full rank, K(W˜ )⫽0 共by continuity this construction is always pos- sible兲. But具␺^⬍k兩W˜兩␺^⬍k典⭓⑀⬎0 since W is a k-SW, ergo no 兩␺^⬍k典苸K(W˜ ). 䊏

Definition 5. A k-class Slater witness W is tangent to Sl_k⫺1 at␳^if᭚a state␳苸Sl_k⫺1 such that Tr(W␳⁾⫽0.

Observation 1. The state ␳is of Slater class k⫺1 iff for all k-SW’s tangent to Sl_k⫺1, Tr(W␳⁾⭓0.

Proof 共see 关10兴兲. 共only if兲 Suppose that ␳ is of class k.

From the Hahn-Banach theorem it follows that there exists a k-SW, W, that detects it. We can subtract⑀1^afrom W, making W⫺⑀1a tangent to Sl_k⫺1 at some ␴, but then Tr关␳^(W

⫺⑀₁⁾兴⬍0.䊏

A. Optimal Slater witnesses

We will now discuss the optimization of a Slater witness.

As proposed in 关10兴 and 关17兴, an entanglement witness 共Schmidt witness兲 W is optimal if there exists no other wit- ness that detects more states than it. The same definition can be applied to Slater witnesses. We say that a k-Slater witness W₂ is finer than a k-Slater witness W₁, if W₂ detects more states than W₁. Analogously, we define a k-Slater witness W to be optimal when there exists no finer witness than itself.

Let us define the set of 兩␺^⬍k典 pure states of Slater rank k

⫺1 for which the expectation value of the k-Slater witness W vanishes:

T_W⫽兵^兩␺^⬍k典 ^{such that} 具␺^⬍k兩W兩␻^⬍k典⫽0其^{, 共27兲} i.e., the set of pure tangent states of Slater rank⬍k. Here W is an optimal k-SW iff W⫺⑀P is not a k-SW, for any posi- tive operator P. If the set T_W spans the whole Hilbert space A(C^2K丢C^2K), then W is an optimal k-SW. If T_W does not spanA(C^2K丢C^2K), then we can optimize the witness by sub- tracting from it a positive operator P, such that PT_W⫽0. For example, for Slater witnesses of class 2 this is possible pro- vided that inf_兩e典苸C2K关P_e^⫺1/2W_eP_e^⫺1/2兴min⬎0. Here for any X acting onA(C^2K丢C^2K), we define

X_e⫽关具^e,•兩X兩e,•典⫺具^e,•兩X兩•,e典⫺具•,e兩X兩e,•典

⫹具•,e兩X兩•,e典兴, 共28兲

as an operator acting inC^2K, and关X兴mindenotes its minimal eigenvalue共see 关10兴兲. An example of an optimal witness of Slater number k inA(C^2K丢C^2K) is given by

W⫽1a⫺ K

k⫺1P, 共29兲

wherePis a projector onto a ‘‘maximally correlated state,’’

兩⌿典⫽(1/

冑

^K)兺i⫽1 K f_a

1(i)

† f_a

2(i)

† 兩⍀典 关cf. Eq. 共5兲兴. The reader can easily check that the above witness operator has mean value zero in the states f_a

1(i)

† f_a

2(i)

† 兩⍀典^{for i}⫽1,2, but also for all states of the form g₁^†g₂^†兩⍀典^{, where}

g₁^†⫽f_a

1共1兲

† e^i␸11⫹f_a

1共2兲

† e^i␸12⫹f_a

2共1兲

† e^i␸21⫹f_a

2共2兲

† e^i␸22, 共30兲 g₂^†⫽⫺f_a

2共1兲

† e^⫺i␸12⫹f_a

1共2兲

† e^⫺i␸11⫺f_a

2共1兲

† e^⫺i␸22

⫹f_a

2共2兲

† e^⫺i␸21, 共31兲

for arbitrary␸i j, i, j⫽1,2. The set T_Wspans in this case the whole Hilbert space A(C^2K丢C^2K): ergo W is optimal.

B. Slater witnesses and positive maps

It is interesting to consider linear maps associated with Slater witnesses via the Jamiołkowski isomorphism 关21兴. Such maps employ W acting in HA丢HB⫽C^2K丢C^2K and transform a state ␳ ^{acting in} HA丢HC⫽C^2K丢C^2K into an- other state acting in HB丢HC⫽C^2K丢C^2K, M (␳⁾

⫽Tr_A(W␳A

T). Obviously, such maps are positive on sepa- rable states: When ␳ is separable, then for any 兩⌿典苸HB 丢H_C, the mean value of具^{⌿兩M (}␳⁾兩⌿典, becomes a convex sum of mean values of W in some product states 兩e, f典 苸HA丢HB. Since W acts in fact in the antisymmetric space, we can antisymmetrize these states, i.e., 兩e, f典→(兩e, f典

⫺兩f ,e典). Such antisymmetric states have, however, Slater rank 1, and all SW’s-of class k⭓2 have thus positive mean value in those states. This class of positive maps is quite different from the ones considered in Refs. 关10兴,关16兴; they provide thus an interesting class of necessary separability conditions. The map associated with the witness 共29兲 is, however, decomposable; i.e., it is a sum of a completely positive map and another completely positive map composed with transposition. This follows from the fact that the witness operator has a positive partial transpose; i.e., it can be pre- sented as a partial transpose of a positive operator.

VI. CONCLUSIONS AND OUTLOOK

Summarizing, we have presented a general characteriza- tion of quantum correlated states in two-fermion systems with a 2K-dimensional single-particle space. This goal has been achieved by introducing the concepts of Slater deom- position and rank for pure states, and Slater number for mixed states. In particular, for the important case K⫽2 the quantum correlations in mixed states can be characterized completely in analogy to Wootters’ result for separated qubits 关18兴and using the findings of Ref.关13兴for pure states. Simi-

larly to the case of separated systems, the situation for K

⬎2 is more complicated. Therefore, we have also introduced witnesses of Slater number k and presented the methods of optimizing them.

Possible directions for future work include generalizations of the present results to more than two fermions and the development of an analogous theory for indistinguishable bosons. For this purpose a lot of the concepts developed so far are expected to be useful there as well. However, there are certainly also fundamental differences between quantum correlations in bosonic and fermionic systems. As an ex- ample, consider the notion of unextendible product bases in- troduced recently for separated systems关26兴. These are sets of product states spanning a subspace of the Hilbert space whose orthogonal complement does not contain any product states. All such unextendible product bases constructed so far involve product states of the form兩␺典^丢兩␹典 ^with兩␺典and兩␹典 being nonorthogonal. In the analogous fermionic state non- orthogonal contributions are obviously cancelled out by an- tisymmetrization, unlike the bosonic case. In fact, all explicit constructions of unextendible product bases known so far 关26兴 can be taken over directly to bosonic systems to give

‘‘unextendible Slater permanent bases.’’ These are sets of symmetrized product states spanning a subspace of the sym- metrized Hilbert space, whose orthogonal complement does not contain any such states.

ACKNOWLEDGMENTS

We thank Anna Sanpera and W. K. Wootters for useful discussions, and Allan H. MacDonald for helpful comments and a critical reading of the manuscript. J.S. was supported by the Deutsche Forschungsgemeinschaft under Grant No.

SCHL 539/1-1. M.K. was supported by Polish KBN Grant No. 2 P03B 072 19. M.L. acknowledges support by the Deutsche Forschungsgemeinschaft via SFB 407, Schwer- punkt ‘‘Quanteninformationsverarbeitung,’’ and the Projekt 436 POL 133/86/0. D.L. acknowledges partial support from the Swiss National Science Foundation.

APPENDIX

We now list further properties of the correlation measure

␩ for pure states 兩⌿典⫽兺a,b⫽1

4 w_abf_a^†f_b^†兩⍀典 of two fermions in a four-dimensional single-particle space 关13兴 and add some further remarks.

The matrix w transforms under a unitary transformation of the one-particle space,

f_a^†哫Uf_a^†U^†⫽

兺

b

U_baf_b^†, 共A1兲 as

w哫UwU^T, 共A2兲

where U^Tis the transpose共not the adjoint兲of U. Under such a transformation, 兩⌿典哫兩⌽典⫽U兩⌿典, scalar products of the form具⌿˜

1兩⌿2典 remain unchanged up to a phase,